:

## incorrect improper integral

Maple 16

This example was reported to me after a Calculus II student encountered this ridiculous result:

f := (k+5)/sqrt(k^7+k^2):
Int( f, k=1..infinity ):
% = value( % );
/infinity
| k + 5
| -------------- dk = -infinity
| (1/2)
/1 / 7 2\
\k + k /

This result is absurd because the integrand is positive, and decays to zero fast enough for this integral to exist.

Let's try to dig a little deeper to how Maple might be coming to this result:

f2 := simplify( f ) assuming positive;
k + 5
---------------
(1/2)
/ 5 \
k \k + 1/
int( f2, k );
/[1 1] [6] 5\
k hypergeom|[-, -], [-], -k |
\[5 2] [5] /

+

/ (1/2)\
(1/2) (1/2) |1 1 / 5 \ |
(-2 ln(2) + 5 ln(k)) Pi - 2 Pi ln|- + - \k + 1/ |
\2 2 /
----------------------------------------------------------------
(1/2)
Pi
qq := int( f2, k=1..N ) assuming positive;
(3/2) /1 \ /1 \
Pi csc|- Pi| sec|- Pi|
\5 / \5 / 2 /[3 1] [13] \
- --------------------------- + - hypergeom|[--, -], [--], -1|
/4\ / 7\ 3 \[10 2] [10] /
5 GAMMA|-| GAMMA|--|
\5/ \10/

/1 1 (1/2)\ /[1 1] [6] 5\
+ 2 ln|- + - 2 | + hypergeom|[-, -], [-], -N | N + 5 ln(N)
\2 2 / \[5 2] [5] /

/ (1/2)\
|1 1 / 5 \ |
- 2 ln|- + - \N + 1/ |
\2 2 /
limit( qq, N=infinity );
/ (3/2) /1 \ /1 \
| Pi csc|- Pi| sec|- Pi|
| \5 / \5 /
limit|- ---------------------------
| /4\ / 7\
| 5 GAMMA|-| GAMMA|--|
\ \5/ \10/

2 /[3 1] [13] \ /1 1 (1/2)\
+ - hypergeom|[--, -], [--], -1| + 2 ln|- + - 2 |
3 \[10 2] [10] / \2 2 /

/[1 1] [6] 5\
+ hypergeom|[-, -], [-], -N | N + 5 ln(N)
\[5 2] [5] /

\
/ (1/2)\ |
|1 1 / 5 \ | |
- 2 ln|- + - \N + 1/ |, N = infinity|
\2 2 / |
|
/
value( % );
/ (3/2) /1 \ /1 \
| Pi csc|- Pi| sec|- Pi|
| \5 / \5 /
limit|- ---------------------------
| /4\ / 7\
| 5 GAMMA|-| GAMMA|--|
\ \5/ \10/

2 /[3 1] [13] \ /1 1 (1/2)\
+ - hypergeom|[--, -], [--], -1| + 2 ln|- + - 2 |
3 \[10 2] [10] / \2 2 /

/[1 1] [6] 5\
+ hypergeom|[-, -], [-], -N | N + 5 ln(N)
\[5 2] [5] /

\
/ (1/2)\ |
|1 1 / 5 \ | |
- 2 ln|- + - \N + 1/ |, N = infinity|
\2 2 / |
|
/
evalf( %% );
0.

This result is equally suspect, for the same reason. There is definitely a positive area under this curve, so the integral can't be zero - it should be either real and positive or infinite.

From a plot of q2 it is pretty clear that the value of this improper integral is about 2.366

This appears to be a bug. Can anyone give an explanation that disputes my conjecture that this is an error?

Doug

---------------------------------------------------------------------